LDA and QDA: Handwritten Digits
Let’s continue with the digits data. We read-in the data:
url <- "https://raw.githubusercontent.com/datasciencelabs/data/master/hand-written-digits-train.csv"
if(!exists("digits")) digits <- read_csv(url)To simplify the problem we will try to distinguish 2s from 7s. So we subset to only those digits
dat <- digits %>% filter(label%in%c(2,7))For illustrative purposes we created two features: X_1 is the percent of black pixels that are in the top left quadrant and X_2 is the percent of black pixels that are in the bottom right quadrant:
We can create these new predictors like this:
dat <- mutate(dat, label = as.character(label)) %>%
mutate(y = ifelse(label == "2",0,1 ))
row_column <- expand.grid(row=1:28, col=1:28)
ind1 <- which(row_column$col <= 14 & row_column$row <= 14)
ind2 <- which(row_column$col > 14 & row_column$row > 14)
ind <- c(ind1,ind2)
X <- as.matrix(dat[,-1])
X <- X > 200
X1 <- rowSums(X[,ind1])/rowSums(X)
X2 <- rowSums(X[,ind2])/rowSums(X)
dat <- mutate(dat, X_1 = X1, X_2 = X2)We can see some examples of what these predictors are:
We act as if we know the truth:
Quadratic and Linear Discriminant Analysis
For illustration purposes let’s take a subset of 1,000 handwritten digits:
set.seed(1)
dat <- sample_n(dat, 1000) %>% select(y, X_1, X_2)
dat## # A tibble: 1,000 x 3
## y X_1 X_2
## <dbl> <dbl> <dbl>
## 1 1 0.238 0.417
## 2 0 0.0610 0.293
## 3 0 0.151 0.315
## 4 0 0.0957 0.255
## 5 0 0.189 0.256
## 6 0 0 0.234
## 7 0 0.151 0.256
## 8 0 0.0192 0.212
## 9 1 0.129 0.186
## 10 0 0.0854 0.207
## # … with 990 more rowsNow create train and test sets:
inTrain <- createDataPartition(y = dat$y, p = 0.5)
train_set <- slice(dat, inTrain$Resample1)
test_set <- slice(dat, -inTrain$Resample1)Quadratic Discriminant Analysis (QDA) relates to the Naive Bayes approach we described earlier. We try to estimate \(\mbox{Pr}(Y=1|X=x)\) using Bayes theorem.
\[ p(x) = \mbox{Pr}(Y=1|\mathbf{X}=\mathbf{x}) = \frac{\pi f_{\mathbf{X}|Y=1}(x)} {(1-\pi) f_{\mathbf{X}|Y=0}(x) + \pi f_{\mathbf{X}|Y=1}(x)} \]
With QDA we assume that the distributions \(f_{\mathbf{X}|Y=1}(x)\) and \(f_{\mathbf{X}|Y=0}(x)\) are multivariate normal. In our case we have two predictors so we assume each one is bivariate normal. This implies we need to estimate two averages, two standard deviations, and a correlation for each case: \(Y=1\) and \(Y=0\).
This implies that we can approximate the distributions \(f_{X_1,X_2|Y=1}\) and \(f_{X_1, X_2|Y=0}\). We can easily estimate parameters from the data:
options(digits = 2)
params <- train_set %>% group_by(y) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2), sd_1= sd(X_1), sd_2 = sd(X_2), r = cor(X_1,X_2))
params## # A tibble: 2 x 6
## y avg_1 avg_2 sd_1 sd_2 r
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0 0.130 0.286 0.0674 0.0592 0.436
## 2 1 0.232 0.292 0.0750 0.107 0.360So here are the data and contour plots showing the two normal densities:
train_set %>% mutate(y = factor(y)) %>%
ggplot(aes(X_1, X_2, fill = y, color = y)) +
geom_point(pch = 21, cex = 5, color = "black") +
stat_ellipse(lwd = 2, type = "norm")
This defines the following estimate of \(f(x_1, x_2)\)
library(mvtnorm)
get_p <- function(params, data){
dmvnorm( cbind(data$X_1, data$X_2),
mean = c(params$avg_1, params$avg_2),
sigma = matrix( c(params$sd_1^2,
params$sd_1*params$sd_2*params$r,
params$sd_1*params$sd_2*params$r,
params$sd_2^2),2,2))
}
pi <- 0.5
p0 <- get_p(params[1,], true_f)
p1 <- get_p(params[2,], true_f)
f_hat_qda <- pi*p1/(pi*p1 + (1-pi)*p0)
p <-true_f %>% mutate(f=f_hat_qda) %>%
ggplot(aes(X_1, X_2, fill=f)) +
scale_fill_gradientn(colors=c("#F8766D","white","#00BFC4")) +
geom_raster() +
stat_contour(aes(x=X_1,y=X_2,z=f),
data=mutate(true_f, f=f_hat_qda),
breaks=c(0.5),color="black",lwd=1.5)
grid.arrange(true_f_plot, p, nrow=1)
Here we have 2 predictors and had to compute 4 means, 4 SDs and 2 correlations. How many parameters would we have if instead of 2 predictors we had 10?
The main problem comes from estimating correlations for 10 predictors. With 10, we have 45 correlations for each class. In general the formula is \(p(p-1)/2\) which gets big quickly.
A solution to this is to assume that the correlation structure is the same for all classes. This reduces the number of parameters we need to estimate. When we do this, we can show mathematically that the solution is “linear”, in the linear algebra sense, and we call it Linear Discriminant Analysis (LDA).
params <- train_set %>% group_by(y) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2), sd_1= sd(X_1), sd_2 = sd(X_2), r = cor(X_1,X_2))
params <- params %>% mutate(sd_1 = mean(sd_1), sd_2=mean(sd_1), r=mean(r))
params ## # A tibble: 2 x 6
## y avg_1 avg_2 sd_1 sd_2 r
## <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0 0.130 0.286 0.0712 0.0712 0.398
## 2 1 0.232 0.292 0.0712 0.0712 0.398This defines the following estimate of \(f(x_1, x_2)\) and the boundary becomes linear:
p0 <- get_p(params[1,], data=true_f)
p1 <- get_p(params[2,], data=true_f)
p <- 0.5
f_hat_lda <- pi*p1/(pi*p1 + (1-pi)*p0)
p <- true_f %>% mutate(f=f_hat_lda) %>%
ggplot(aes(X_1, X_2, fill=f)) +
scale_fill_gradientn(colors=c("#F8766D","white","#00BFC4")) +
geom_raster() +
stat_contour(aes(x=X_1,y=X_2,z=f),
data=mutate(true_f, f=f_hat_lda),
breaks=c(0.5),color="black",lwd=1.5)
grid.arrange(true_f_plot, p, nrow=1)
ROC
With the example we have been examining we can make two types of errors: calling a 2 a 7 or calling a 7 a 2. More generally, with binary data we call these false positives (calling a 0 a 1) and false negatives (calling a 1 a 0). Here we have arbitrarily made 7s 1s and 2s 0s.
This concept is important in many areas and in particular in health where one type of mistake can be much more costly than another. Note that we have been predicting 1s based on the rule \(\hat{f}(x_1, x_2) > 0.5\) but we can pick another cutoff, depending on the cost of each type of error. For example, if we are predicting if a plane will malfunction, then we want a very low false negative rate and are willing to sacrifice our true positive rate.
We can see that the estimated probabilities are on a continuum:
params <- train_set %>% group_by(y) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2),
sd_1= sd(X_1), sd_2 = sd(X_2),
r = cor(X_1,X_2))
p0 <- get_p(params[1,], dat= test_set)
p1 <- get_p(params[2,], dat= test_set)
pi <- 0.5
pred_qda <- pi*p1/(pi*p1 + (1-pi)*p0)
test_set %>% mutate(pred=pred_qda, label=as.factor(y)) %>%
ggplot(aes(label,pred)) + geom_boxplot()
The Receiver Operator Characteristic Curve (ROC Curve) plots true positive rate versus false positive rate for several choices of cutoff. We can create this curve with the following code:
library(pROC)
roc_qda <- roc(test_set$y, pred_qda)
plot(roc_qda)
Here are the results for LDA
params <-params %>% mutate(sd_1 = mean(sd_1),
sd_2 = mean(sd_1),
r=mean(r))
p0 <- get_p(params[1,], dat = test_set)
p1 <- get_p(params[2,], dat = test_set)
pi <- 0.5
pred_lda <- pi*p1/(pi*p1 + (1-pi)*p0)
roc_lda <- roc(test_set$y, pred_lda)## Setting levels: control = 0, case = 1## Setting direction: controls < casesplot(roc_qda)
plot(roc_lda, add=TRUE, col=2)
legend(0, 0.5, legend=c("QDA", "LDA"),
col=c("black", "red"), lty=1)
We can also compare to KNN with two different values of k: 5 and 51.
fit <- knn3(y~., data = train_set, k = 5)
pred_knn_5 <- predict(fit, newdata = test_set)[,2]
roc_knn_5 <- roc(test_set$y, pred_knn_5)## Setting levels: control = 0, case = 1## Setting direction: controls < casesplot(roc_qda)
plot(roc_knn_5, add=TRUE, col=3)
legend(0, 0.5, legend=c("QDA", "KNN, k = 5"),
col=c("black", 3), lty=1)
fit <- knn3(y~., data = train_set, k = 51)
pred_knn_51 <- predict(fit, newdata = test_set)[,2]
roc_knn_51 <- roc(test_set$y, pred_knn_51)## Setting levels: control = 0, case = 1
## Setting direction: controls < casesplot(roc_qda)
plot(roc_knn_51, add=TRUE, col=4)
legend(0, 0.5, legend=c("QDA", "KNN, k = 51"),
col=c("black", 4), lty=1)
To summarize these curves into one single number that can be compared across methods, it is common to take the area under the ROC curve (abbreviated AUC or AUROC). Higher values indicate better performance.
auc(roc_qda)## Area under the curve: 0.92auc(roc_lda)## Area under the curve: 0.89auc(roc_knn_5)## Area under the curve: 0.88auc(roc_knn_51)## Area under the curve: 0.92Three classes
Here we look at an example where there are three classes to consider. In addition to 2s and 7s, we also consider 1s. Now we need to estimate three different curves that represent the conditional probabilities of being each digit given the values of \(X_1\) and \(X_2\).
We’ll create a training set and test set so we can test out how each method does on the three class problem.
set.seed(1)
dat <- sample_n(dat, 3000) %>% select(label, X_1, X_2) %>% mutate(label = as.factor(label))
inTrain <- createDataPartition(y = dat$label, p = 0.5)
train_set <- slice(dat, inTrain$Resample1)
test_set <- slice(dat, -inTrain$Resample1)train_set %>% ggplot(aes(X_1,X_2,fill=label)) + geom_point(cex=3, pch=21)
LDA on three classes
params <- train_set %>% group_by(label) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2), sd_1= sd(X_1), sd_2 = sd(X_2), r = cor(X_1,X_2))
params <-params %>% mutate(sd_1 = mean(sd_1), sd_2=mean(sd_1), r=mean(r))
params ## # A tibble: 3 x 6
## label avg_1 avg_2 sd_1 sd_2 r
## <fct> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 0.0708 0.246 0.0826 0.0826 0.436
## 2 2 0.132 0.280 0.0826 0.0826 0.436
## 3 7 0.240 0.292 0.0826 0.0826 0.436This defines the following estimate of \(f(x_1, x_2)\) and the boundary becomes linear:
p0 <- get_p(params[1,], true_f)
p1 <- get_p(params[2,], true_f)
p2 <- get_p(params[3,], true_f)
pred <- apply(cbind(p0, p1, p2),1,which.max)
tmp <- true_f %>% mutate(pred=pred)
tmp %>% ggplot() +
stat_contour(aes(x=X_1,y=X_2,z=pred),
breaks=c(1,2,3),color="black",lwd=1.5) +
geom_point(aes(X_1,X_2,fill=label), dat=test_set,pch=21)
QDA on three classes
params <- train_set %>% group_by(label) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2), sd_1= sd(X_1), sd_2 = sd(X_2), r = cor(X_1,X_2))This defines the following estimate of \(f(x_1, x_2)\) and the boundary becomes non-linear:
p0 <- get_p(params[1,], true_f)
p1 <- get_p(params[2,], true_f)
p2 <- get_p(params[3,], true_f)
pred <- apply(cbind(p0, p1, p2),1,which.max)
tmp <- true_f %>% mutate(pred=pred)
tmp %>% ggplot() +
stat_contour(aes(x=X_1,y=X_2,z=pred),
breaks=c(1,2,3),color="black",lwd=1.5) +
geom_point(aes(X_1,X_2,fill=label), dat=test_set,pch=21)
GLM on three classes
Let’s compare the performance of LDA and QDA with logistic regression. For each label, we will train a model to predict that number, or not that number. So fit1 predicts 1 vs not 1, fit2 predicts 2 vs not 2, and fit7 predicts 7 vs not 7. Then we can also plot these boundaries.
fit1 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="1"),family="binomial")
fit2 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="2"),family="binomial")
fit7 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="7"),family="binomial")
f_hat1 <- predict(fit1, newdata = true_f, type = "response")
f_hat2 <- predict(fit2, newdata = true_f, type ="response")
f_hat7 <- predict(fit7, newdata = true_f, type = "response")
pred <- apply(cbind(f_hat1, f_hat2, f_hat7),1,which.max)
tmp <- true_f %>% mutate(pred=pred)
tmp %>% ggplot() +
stat_contour(aes(x=X_1,y=X_2,z=pred),
breaks=c(1,2,3),color="black",lwd=1.5) +
geom_point(aes(X_1,X_2,fill=label), dat=test_set,pch=21)
kNN on three classes
Let’s also use kNN with k = 51 and k = 101 and draw the boundaries.
fit <- knn3(label~., data=train_set, k=51)
f_hat <- predict(fit, newdata = true_f)
f_hat_max <- apply(f_hat,1,max)
pred <- apply(f_hat,1,which.max)
tmp <- true_f %>% mutate(pred=pred)
tmp %>% ggplot() +
stat_contour(aes(x=X_1,y=X_2,z=pred),
breaks=c(1,2,3),color="black",lwd=1.5) +
geom_point(aes(X_1,X_2,fill=label), dat=test_set,pch=21)
fit <- knn3(label~., data=train_set, k=101)
f_hat <- predict(fit, newdata = true_f)
f_hat_max <- apply(f_hat,1,max)
pred <- apply(f_hat,1,which.max)
tmp <- true_f %>% mutate(pred=pred)
tmp %>% ggplot() +
stat_contour(aes(x=X_1,y=X_2,z=pred),
breaks=c(1,2,3),color="black",lwd=1.5) +
geom_point(aes(X_1,X_2,fill=label), dat=test_set,pch=21)
Comparison
After training each model we can predict lables for the test set and then compare the accuracy of each method.
##QDA
params <- train_set %>% group_by(label) %>%
summarize(avg_1 = mean(X_1), avg_2 = mean(X_2), sd_1= sd(X_1), sd_2 = sd(X_2), r = cor(X_1,X_2))
p0 <- get_p(params[1,], test_set)
p1 <- get_p(params[2,], test_set)
p2 <- get_p(params[3,], test_set)
pred_qda <- apply(cbind(p0, p1, p2),1,which.max)
##LDA
params <-params %>% mutate(sd_1 = mean(sd_1), sd_2=mean(sd_1), r=mean(r))
p0 <- get_p(params[1,], test_set)
p1 <- get_p(params[2,], test_set)
p2 <- get_p(params[3,], test_set)
pred_lda <- apply(cbind(p0, p1, p2),1,which.max)
##GLM
fit1 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="1"),family="binomial")
fit2 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="2"),family="binomial")
fit7 <- glm(y~X_1+X_2, data=mutate(train_set,
y=label=="7"),family="binomial")
f_hat1 <- predict(fit1, newdata = test_set, type = "response")
f_hat2 <- predict(fit2, newdata = test_set, type ="response")
f_hat7 <- predict(fit7, newdata = test_set, type = "response")
pred_glm <- apply(cbind(f_hat1, f_hat2, f_hat7),1,which.max)
##KNN 51
fit <- knn3(label~., data=train_set, k=51)
f_hat <- predict(fit, newdata = test_set)
pred_knn_51 <- apply(f_hat,1,which.max)
##KNN 101
fit <- knn3(label~., data=train_set, k=101)
f_hat <- predict(fit, newdata = test_set)
pred_knn_101 <- apply(f_hat,1,which.max)Let’s compare:
tab <- table(factor(pred_lda, labels=c("1","2","7")), test_set$label)
confusionMatrix(tab)## Confusion Matrix and Statistics
##
##
## 1 2 7
## 1 393 155 26
## 2 67 210 89
## 7 71 97 391
##
## Overall Statistics
##
## Accuracy : 0.663
## 95% CI : (0.639, 0.687)
## No Information Rate : 0.354
## P-Value [Acc > NIR] : < 2e-16
##
## Kappa : 0.492
##
## Mcnemar's Test P-Value : 3.99e-12
##
## Statistics by Class:
##
## Class: 1 Class: 2 Class: 7
## Sensitivity 0.740 0.455 0.773
## Specificity 0.813 0.850 0.831
## Pos Pred Value 0.685 0.574 0.699
## Neg Pred Value 0.851 0.778 0.878
## Prevalence 0.354 0.308 0.338
## Detection Rate 0.262 0.140 0.261
## Detection Prevalence 0.383 0.244 0.373
## Balanced Accuracy 0.777 0.652 0.802confusionMatrix(tab)$overall[1]## Accuracy
## 0.66tab <- table(factor(pred_qda, labels=c("1","2","7")), test_set$label)
confusionMatrix(tab)$overall[1]## Accuracy
## 0.74tab <- table(factor(pred_glm, labels=c("1","2","7")), test_set$label)
confusionMatrix(tab)$overall[1]## Accuracy
## 0.61tab <- table(factor(pred_knn_51, labels=c("1","2","7")), test_set$label)
confusionMatrix(tab)$overall[1]## Accuracy
## 0.75tab <- table(factor(pred_knn_101, labels=c("1","2","7")), test_set$label)
confusionMatrix(tab)$overall[1]## Accuracy
## 0.74